Mathematical models in population dynamics and ecology

Transcription

1 Mathematical models in population dynamics and ecology Rui DILÃO Institut des Hautes Études Scientifiques 35, route de Chartres Bures-sur-Yvette (France) Juin 2004 IHES/M/04/26

2 CHAPTER 15 MATHEMATICAL MODELS IN POPULATION DYNAMICS AND ECOLOGY Rui Dilão Non-Linear Dynamics Group, Instituto Superior Técnico Av. Rovisco Pais, Lisbon, Portugal and Institut des Hautes Études Scientifiques Le Bois-Marie, 35, route de Chartres, F Bures-sur-Yvette, France We introduce the most common quantitative approaches to population dynamics and ecology, emphasizing the different theoretical foundations and assumptions. These populations can be aggregates of cells, simple unicellular organisms, plants or animals. The basic types of biological interactions are analysed: consumer-resource, prey-predation, competition and mutualism. Some of the modern developments associated with the concepts of chaos, quasi-periodicity, and structural stability are discussed. To describe short- and long-range population dispersal, the integral equation approach is derived, and some of its consequences are analysed. We derive the standard McKendrick age-structured density dependent model, and a particular solution of the McKendrick equation is obtained by elementary methods. The existence of demography growth cycles is discussed, and the differences between mitotic and sexual reproduction types are analysed. Contents 1 Introduction Biotic Interactions One species interaction with the environment Two interacting species Discrete models for single populations. Age-structured models A case study with a simple linear discrete model Discrete time models with population dependent parameters Resource dependent discrete models Spatial effects Age-structured density dependent models Growth by mitosis Conclusions References

3 2 R. Dilão 1. Introduction In a region or territory, the number of individuals of a species or a community of species changes along the time. This variation is due to the mechanisms of reproduction and to the physiology of individuals, to the resources supplied by the environment and to the interactions or absence of interactions between individuals of the same or of different species. Biology is concerned with the architecture of living organisms, its physiology and the mechanisms that originated life from the natural elements. Ecology studies the relations between living organisms and the environment, and, in a first approach, detailed physiological mechanisms of individuals have a secondary importance. As a whole, understanding the phenomena of life and the interplay between living systems and the environment make biological sciences, biology together with ecology, a complex science. To handle the difficulties inherently associated to the study of living systems, the input of chemistry, physics and mathematics is fundamental for the development of an integrative view of life phenomena. In the quantitative description of the growth of a population, several interactions are involved. There are intrinsic interactions between each organism and its environment and biotic interactions between individuals of the same or of different species. These interactions have specific characteristic times or time scales, and affect the growth and fate of a species or a community of species. Population dynamics deals with the population growth within a short time scale, where evolutionary changes and mutations do not affect significantly the growth of the population, and the population is physiologically stable. In this short time scale, the variation of the number of individuals in the population is determined by reproduction and death rates, food supply, climate changes and biotic interactions, like predation, competition, mutualism, parasitism, disease and social context. There exists an intrinsic difficulty in analysing the factors influencing the growth and death of a species. There are species that are in the middle of a trophic web, being simultaneously preys and predators, and the trophic web exhibits a large number of interactions. For example, the food web of Little Rock Lake, Wisconsin, shows thousands of inter-specific connections between the top levels predators down to the phytoplankton, [1]. In this context, organisms in batch cultures and the human population are the simplest populations. In batch cultures, organisms interact with their resources for reproduction and growth. The human population is at the top of a trophic chain. Even for each of these simple cases, we can have different modelling approaches and strategies. From the observational point of view, one of the best-known populations is the human population for which we have more than 50 years of relatively accurate census, and some estimates of population numbers over larger time intervals. Observations of population growth of micro-organisms in batch cultures are important to validate models and to test growth projections based on mathematical models. Mathematical models give an important contribution to ecological studies. They

4 Mathematical Models in Population Dynamics and Ecology 3 propose quantities that can be measured, define concepts enabling to quantify biological interactions, and even propose different modelling strategies with different assumptions to describe particular features of the populations. In population dynamics, and from the mathematical point of view, there are essentially two major modelling strategies: i) The continuous time approach using techniques of ordinary differential equations; ii) The discrete time approach which is more closely related with the structure of the census of a population. Both approaches use extensively techniques of the qualitative theory of dynamical systems. In the continuous time approach, the number of individuals of a population varies continuously in time and the most common modelling framework applies to the description of the types of biotic inter-specific interactions and to the interactions of one species with the environment. They are useful for the determination of the fate of a single population or of a small number of interacting species. These models have been pioneered by Pierre-Françoise Verhulst, in the 19th century, with the introduction of the logistic model, and by Vito Volterra, in the first quarter of the 20th century, with the introduction of a model to describe qualitatively the cycling behaviour of communities of carnivore and herbivore fishes. In the discrete time approach, models are built in order to describe the census data of populations. They are discontinuous in time, and are closer to the way population growth data are obtained. These models are useful for short time prediction, and their parameters can be easily estimated from census data. Modern ecology relies strongly on the concepts of carrying capacity (of the environment) and growth rate of a population, introduced by the discrete and the continuous models. In the 20th century, the works of McKendrick and Leslie gave an important contribution to modern ecology and demography. The usefulness of population dynamics to predictability and resource management depends on the underlying assumptions of the theoretical models. Our goal here is to introduce in a single text the most common quantitative approaches to population dynamics, emphasizing the different theoretical foundations and assumptions. In the next two sections, we introduce the continuous and the discrete agestructured approach to quantitative ecology. These are essentially two review sections, where we emphasise on the assumptions made in the derivation of the models, and whenever possible, we present case studies taken from real data. As the reader has not necessarily a background on the techniques of the qualitative theory of dynamical systems, we introduce some of its geometric tools and the concept of structural stability. In modelling situations where there exists some arbitrariness, structural stability is a useful tool to infer about the qualitative aspects of the solutions of ordinary differential equations upon small variations of its functional forms. In section 4, and in the sequence of the Leslie type age-structured discrete models (Sec. 3), we make the mathematical analysis of the Portuguese population based

5 4 R. Dilão on the census data for the second half of the 20th century. Here we introduce a very simple model in order to interpret data and make demographic projections, to analyse migrations and the change of socio-economic factors. This is a very simple example that shows the importance of mathematical modelling and analysis in population studies. In section 5 we introduce discrete time models with population dependent growth rates, and we analyse the phenomenon of chaos. In section 6, the consumer-resources interaction is introduced, and we discuss the two types of randomness found in dynamical systems: quasi-periodicity and chaos. In section 7, we derive a general approach to the study of population dispersal (short- and longrange), and we derive a simple integro-difference equation to analyse the dispersion of a population. In section 8, we introduce the standard continuous model for age-structured density dependent populations, the McKendrick model, showing the existence of time periodic solutions by elementary techniques. In this context, we discuss demography cycles and the concept of growth rate. In section 9, we derive a modified McKendrick model for populations with mitotic type reproduction, and compare the growth rates between populations with sexual and mitotic reproduction types. In the final section, we resume the main conclusions derived along the text and we compare the different properties of the analysed models. 2. Biotic Interactions 2.1. One species interaction with the environment We consider a population of a single species in a territory with a well-defined boundary. Let x(t) be the number of individuals at some time t. The growth rate of the population (by individual) is, 1 dx x dt = r (1) If the growth rate r is a constant, independently of the number of individuals of the population, then equation (1) has the exponential solution x(t) =x(t 0 )e r(t t0), where x(t 0 ) is the number of individuals in the population at time t 0. If, r>0, x(t), ast. In a realistic situation, such a population will exhaust resources and will die out in finite time. Equation (1) with r constant is the Malthusian law of population growth. Exponential growth is in general observed in batch cultures of micro-organisms with a large amount of available resources and fast reproduction times, [2] and [3]. From the solution of equation (1) follows that the doubling time of the initial population (t d ) is related with the growth rate by t d =ln2/r. For example, with the data of the world population, [4], we can determine the variation of the doubling time or the growth rate of the human population along historical times, Fig. 1. The curve in Fig. 1 suggests that, for human populations and at a large time scale, the growth rate r cannot be taken constant as in the Malthusian growth law (1), but must depend on other factors, as, for example, large-scale diseases, migrations, etc..

6 Mathematical Models in Population Dynamics and Ecology 5 Fig. 1. Evolution of the doubling time of the world population. The doubling time has been calculated according to the formula t d =ln2/r, where r =(N(t + h) N(t))/(hN(t)), and N(t) is the world population at year t. The data set is from reference [4]. For large population densities, and in order to avoid unrealistic situations of exponential growth or explosion of population numbers, it is expected that the growth rate becomes population dependent. Assuming that, for large population numbers, r r(x(t)) < 0, for x>k, and, r r(x(t)) > 0, for x<k, where K is some arbitrary constant, the simplest form for the growth rate r(x) is, r(x) = r 0 (K x). Substitution of this population dependent growth rate into equation (1) gives, dx dt = r 0x(K x) :=rx(1 x/k) (2) where r 0 is a rate constant. Equation (2) is the logistic or Verhulst equation for onespecies populations. For a population with x(t 0 ) > 0 at some time t 0, the general solution of (2) is, x(t) = x(t 0 )Ke r0(t t0) x(t 0 )(e r0 (t t0) 1) + K and, in the limit t, x(t) K. The constant K is called the carrying capacity of the environment and is defined as the maximum number of individuals of a species that the territory can support. For the same species, larger territories and bigger renewable resources correspond to larger values of K. The logistic equation (2) describes qualitatively the growth of single colonies of micro-organisms in batch experiments, [3]. For example, in a batch experiment, Gause, [5], fitted the measured growth curve of the protozoa Paramecium caudatum, finding a good agreement with the solution (3) of the logistic equation. For the human population, the agreement is not so good, being dependent on technological developments, sociological trends and other factors, [2]. Depending on the data set, and from country to country, some authors find a good fit between the solutions of the logistic equation and demography data (see for example [4]), and others propose (3)

7 6 R. Dilão empirical models based on the delayed logistic equation, x (t) =rx α (1 x(t T )/K), [4]. In the derivation of the logistic equation, the plausibility of the mathematical form of the growth rate is assumed, without any assumptions about the relations between population growth and environmental support, or about the mechanisms of interaction between individuals and the environment. It is simply supposed that, for each species, the environment ensures enough resources. The carrying capacity constant can only be measured a posteriori through the asymptotic solution, x(t) K, ast. A possible mechanism for the derivation of the logistic equation is based on the mass action law of chemical kinetics, [6] and [7, p ]. To be more specific, we represent species and resources by, A j, with j =1,...,m. The interactions between species or between species and resources can be represented by n collision diagrams, ν i1 A ν im A m r i µi1 A µ im A m, i =1,...,n (4) where r i measure the rate at which the interactions occur, and the constants ν ij and µ ij are positive parameters measuring the number of individuals or units of resources that are consumed or produced in a collision. The mass action law asserts that the time evolution of the (mean) concentration of A j is given by, da j dt = n i=1 r i (µ ij ν ij )A νi1 1...A νim m, j =1,...,m (5) As we have in general n interaction diagrams and m species or resources, the system of equations (5) are not independent. In general, by simple inspection of the m equations (5), it is possible to derive the associated conservation laws, that is, a set of linear relations between the concentrations A j. With these conservations laws, we obtain a system of s m linearly independent differential equations. In this framework, reproduction in the presence of resources can be seen as the collision of the members of a population with the resources. In the case of the logistic equation, the collisions between individuals and the resource is represented by the diagram, A + x r0 (1 + e)x (6) where A represents resources, x is the number of individuals in the population, collisions occur at the rate r 0, and the inequality e>0 expresses the increase in the number of individuals. By (4) and (5), to the diagram (6) is associated the logistic equation (2), together with the conservation law x(t) +ea(t) = x(t 0 )+ea(t 0 ), where the carrying capacity is given by K = x(t 0 )+ea(t 0 ). As, in the limit t, x(t) K, then, in the same limit, A(t) 0. In this interpretative framework, when the population attains the equilibrium value K, resources are exhausted. In realistic situations, after reaching the equilibrium, the number of individuals of the population decreases and, some time afterwards, the population disappears, [3]. However, this asymptotic behaviour is not predicted by equation (2). To further

8 Mathematical Models in Population Dynamics and Ecology 7 include this effect, we can add to the collision mechanism (6) a new death rate diagram, x. d In this case, by the mass action law, (4) and (5), the time variation of the number of individuals of the population is not of logistic type anymore, obeying to the equations, { x (t) =r 0 exa dx A (7) (t) = r 0 xa without any conservation law and, consequently, without a carrying capacity parameter. Numerical integration of the system of equations (7) leads to the conclusion that, for a small initial population, a fast exponential growing phase is followed by a decrease in the number of individuals of the population, and extinction occurs when t, Fig. 2a). The growth behaviour predicted by equation (7) is in qualitative agreement with the growth curves observed in generic microbiological batch experiments, [3]. In Fig. 2a), we compare the solutions of the three equations (1), (2) and (7) for the growth of one-species. In the growing phase, the solutions of the three growth models show qualitatively the same type of exponential behaviour. For equation (7), the concept of carrying capacity is lost but the growth maximum is approximated by the value of the carrying capacity of the logistic equation. In these models, and for the same data set, it is possible to obtain different values for the fitted growth rates, as it is clearly seen in Fig. 2b). Fig. 2. a) Comparison between the solutions of the exponential (1), logistic (2) and equation (7), for the initial conditions x(0) = 1, A(0) = 9, and the parameters r =1,r 0 =1,K = 10, d =0.01 and e = 1. b) Growth rates as a function of time for equations (1), (2) and (7). The approach developed so far introduces into the language of population dynamics the concepts of exponential or Malthusian growth, growth rate, doubling time and carrying capacity. The agreement between the models and data from laboratory experiments is, in some situations, very good, but in others deviates from observations. In the situations where no agreement with observations is found, it is believed that other relevant factors besides reproduction are not included in the modelling process. In modern ecology, the modelling concepts introduced here en-

9 8 R. Dilão able a rough estimate of the population growth and are the starting point for more specific and specialized approaches. For a more extensive study and applications of the exponential and logistic models see [8], [9] and [10] Two interacting species Here, we introduce the basic models for the different types of biotic interactions between the populations of two different species. As models become non-linear, and no general methods for the determination of solutions of non-linear differential equations exist, in parallel, we introduce some of the techniques of the qualitative theory of differential equations (dynamical systems theory). We consider two interacting species in the same territory, and we denote by x(t) and y(t), their total population numbers at time t. The growth rates by individual of both interacting species are, { 1x dx dt = f(x, y) 1 y dy = g(x, y) dt defining the two-dimensional system of differential equations, or vector field, { dx = xf(x, y) dt dy dt = yg(x, y) (8) The particular form of the system of equations (8) ensures that the coordinate axes of the (x, y) phase space are invariant for the flow defined by the vector field (8), in the sense that, any initial condition within any one of the coordinate axis is transported by the phase flow along that axis. Due to this particular invariant property, in the literature of ecology, equations (8) are said to have the Kolmogoroff form, [8, p. 62]. In general, the system of differential equations (8) is non-linear and there are no general methods to integrate it explicitly. We can overcome this problem by looking at equation (8) as defining a flow or vector field in the first quadrant of the two-dimensional phase space (x 0,y 0). Adopting this point of view, the flow lines are the images of the solutions of the differential equation in the phase space, Fig. 3. At each point in phase space, the flow lines have a tangent vector whose coordinates are xf(x, y) and yg(x, y), and the flow lines can be visualised through the graph of the vector field. In fact, given a set of points in phase space, we can calculate the x- and y-coordinates of the vector field components, xf(x, y) and yg(x, y), and draw the directions of the tangent vectors to the flow lines. The solutions of the differential equation (8) are tangent to the vector field. The phase space points for which we have simultaneously, xf(x, y) = 0 and yg(x, y) = 0, are the fixed points of the flow. The fixed points are stationary solutions of the ordinary differential equation (8). In dimension two, the knowledge of these stationary solutions determines the overall topology of the flow lines in

10 Mathematical Models in Population Dynamics and Ecology 9 phase space. With the additional knowledge of the two nullclines, defined by equations xf(x, y) = 0 and yg(x, y) = 0, we can qualitatively draw in phase space the flow lines of the differential equation and to determine the asymptotic states of the dynamics, which, in generic cases, are isolated fixed points. Fig. 3. The differential equation (8) defines a vector field or phase flow in the two-dimensional phase space. The flow lines are the images in phase space of the solutions of the differential equation. The flow lines are parameterised by the time t. At each point (x, y) in phase space, the tangent vector to the flow line or orbit has local coordinates xf(x, y) and yg(x, y). The (isolated) fixed points of a differential equation can be (Lyapunov-) stable or unstable. They are stable if, for any initial condition close to the fixed point, and for each t>0, the solution of the equation remain at a finite distance from the fixed point. If in addition, in the limit t, the solution converges to the fixed point, we say that the fixed point is asymptotically stable. A fixed point is unstable if it is not stable. Around a fixed point, the stability properties of the solutions of a differential equation can be easily analysed. Let (x,y ) be a fixed point of equation (8), and let (x(t) =x + x(t), y(t) =y +ȳ(t)) be a solution defined locally around (x,y ). Introducing this solution into (8), we obtain, up to the first order in x and ȳ, ( dx dt dy dt ) = f(x,y )+x f x (x,y ) y g x (x,y ) x f y (x,y ) g(x,y )+y g y (x,y ) ( ) x =: DF y ( ) x y (9) where DF is the Jacobian matrix of the vector field (8) evaluated at (x,y ). In the conditions of the theorem below, the solutions of the linear differential equation (9) are equivalent to the solutions of the nonlinear equation (8) near (x,y ). Theorem 1: (Hartman-Grobman, [11]) If none of the eigenvalues of the Jacobian matrix DF rest on the imaginary axis of the complex plane, then, near the fixed point (x,y ), the phase flows of equations (8) and (9) are similar or topologically equivalent.

11 10 R. Dilão Under the conditions of the Hartman-Grobman theorem (Theorem 1), by a simple linear analysis, it is possible to determine the stability of the fixed points of the non-linear equation (8), and, therefore, to determine the asymptotic behaviour of the solutions of the non-linear equation (8). The global flow in the first quadrant of phase space is conditioned by the fixed points with non-negative coordinates. This approach is geometrically intuitive and is one of the most powerful tools of the theory of dynamical systems, [11] and [12]. As will see now, this enables the analysis of models for biotic interactions with a minimum of technicalities. We now introduce the most common types of two-species interactions. There are essentially three basic two-species interactions: prey-predator, competition and mutualism. In the prey-predator interaction, for large predator numbers, the growth rate of the prey becomes negative, but in the absence of predators, the growth rate of the prey is positive. If the prey is not the only resource for predators, the growth rate of the predators is always positive. In competition, and in the presence of both species, both growth rates decrease. In mutualistic interactions, the growth rates of both species increase. Adopting the same empirical formalism as in the case of the logistic equation (Sec. 2.1), we assume that the growth rates f and g are sufficiently well behaved functions, and the above ecology definitions can be stated into the mathematical form: f g Prey-predator: < 0, y x > 0 f g Competition: < 0, (10) Mutualism : y f y x < 0 g > 0, x > 0 In the simplest situation where f and g are affine functions, f(x, y) =d 1 +d 2 x+d 3 y and g(x, y) =d 4 + d 5 x + d 6 y, and further assuming that in the absence of one of the species the growth of the other species is of logistic type, by (10), we obtain for the growth rates, Prey-predator: f = r x (1 x/k x c 1 y) and g = r y (1 + c 2 x y/k y ) Competition: f = r x (1 x/k x c 1 y) and g = r y (1 c 2 x y/k y ) (11) Mutualism: f = r x (1 x/k x + c 1 y) and g = r y (1 + c 2 x y/k y ) where c 1, c 2, K x and K y are positive constants. The constants in the growth rate functional forms (11) have been chosen in such a way that, in the absence of any one of the species, we obtain the logistic equation (2). Introducing (11) into (8), we obtain three systems of non-linear ordinary differential equations for prey-predation, competition and mutualism. The topological structure in phase space of the solutions of these equations can be easily analysed by the qualitative methods just described above. The generic differential equation (8) defines a flow in the first quadrant of the two-dimensional phase space, and the simplest solutions are the fixed points of the flow. These fixed points are obtained by solving simultaneously the equations,

12 Mathematical Models in Population Dynamics and Ecology 11 xf(x, y) = 0 and yg(x, y) = 0. For any of the values of the parameters in (11), and for the three biotic interaction types, we have the fixed points (x 0,y 0 )=(0, 0), (x 1,y 1 )=(K x, 0) and (x 2,y 2 )=(0,K y ), which correspond to the absence of one or both species. The fixed points (K x, 0) and (0,K y ) are the asymptotic solutions associated to any non-zero initial condition on the phase space axis x and y, respectively. The zero fixed point corresponds to the absence of both species. For a particular choice of the parameters, a forth fixed point can exists: 1 c 1K y 1+c Prey-predator: (x 3,y 3 )=(K x 1+c 1c 2K xk y,k 2K x y 1+c 1c 2K xk y )ifc 1 K y < 1 c Competition: (x 3,y 3 )=(K 1K y 1 x c,k c 2K x 1 1c 2K xk y 1 y c )ifc 1c 2K xk y 1 1K y > 1, c 2 K x > 1 1+c Mutualism: (x 3,y 3 )=(K 1K y 1+c x 1 c 1c 2K xk y,k 2K x y 1 c 1c 2K xk y )ifc 1 c 2 K x K y < 1 (12) In Fig. 4, we show, for the differential equation (8) and the three growth rate functions (11), all the qualitative structures of the flows in phase space. The fixed points with non-zero coordinates (12) correspond to cases a)-c), and are marked with a square. To determine qualitatively the structure of the solutions of equation (8) for the different cases depicted in Fig. 4, we have analysed the signs of the components of the vector field along the nullclines. The arrows in Fig. 4 represent the directions of the flow in phase space. Except the case of Fig. 4a), the vector field directs the flow towards the fixed points, and the limiting behaviour of the solutions as t is easily derived. To analyse the prey-predator case of Fig. 4a), we have calculated the directions of the vector field near the fixed point (x 3,y 3 ), Fig. 5. In this case, the flow turns around the fixed point (x 3,y 3 ), and to determine the local structure of the flow, we use the technique provided by the Hartman-Grobman theorem. Linearising equation (8) around (x 3,y 3 ), by (11) and (12), we obtain the linear system of differential equations, ( ) dx ( )( ) ( ) dt rx x 3 /K x r x x 3 c 1 K x x x dy = = DF (13) r y c 2 y 3 r y y 3 /K y y y dt where (x, y) =(x 3 + x, y 3 + y). The stability near the fixed point (x 3,y 3 ) is determined by the eigenvalues of the matrix DF, provided that they are not on the imaginary axis of the complex plane. As, T race(df) = λ 1 + λ 2 < 0 and Det(DF) =λ 1 λ 2 > 0, the eigenvalues λ 1 and λ 2 of DF are both real and negative or, complex conjugate with negative real parts. As the solution of the linear system of equations (13) is a linear combination of terms of the form e λit, and the eigenvalues have negative real parts, this implies that x(t) and y(t) converge to zero as t. Therefore, for non-zero initial conditions, the solutions of the prey-predator system of Fig. 4a) converge to the stable fixed point (x 3,y 3 ), Fig. 5. In the prey-predator case of Fig. 4d), c 1 K y > 1, the effect of the predator on the prey is so strong that asymptotically predators consume all the preys, and, as

13 12 R. Dilão Fig. 4. Qualitative structures of the flow in phase space of the differential equation (8), for the growth rate functions (11). Bullets and squares represent fixed points. In cases a)-c), a nonzero fixed point exists if the conditions in (12) are verified. Cases d)-g) correspond to different arrangements of nullclines. The arrows represent the directions of the vector fields, and the solutions of equation (8) are tangent to the vector field. The sign of the vector field is calculated from the sign of the functions f and g at each point in phase space. t, the solutions converge for the fixed point (x 2,y 2 )=(0,K y ). In this case, we do not need to make the linear analysis near the fixed points because the directions of the vector field show clearly the convergence of the solutions to the asymptotically stable fixed point. For the competitive and mutualistic interactions of Figs. 4b) and 4c), equation (8) has always a stable fixed point which is also an asymptotic solution for non-zero initial conditions. In cases e) and g), we have c 1 K y < 1orc 2 K x < 1, and, asymptotically in time, only one of the species survives. For the mutualistic interaction f), we

14 Mathematical Models in Population Dynamics and Ecology 13 Fig. 5. Vector field and nullclines for the prey-predator equation of Fig. 4a), with parameter values c 1 =0.05, c 2 =0.01, K x = 15, K y = 10 and r x = r y = 1. As it is clearly seen, the vector field directs the flow to the fixed point (x 3,y 3 ). This fixed point is asymptotically stable. have c 1 c 2 K x K y > 1, and, asymptotically in time, both population numbers explode to infinity. (Note however that, in this last case, it is possible that the solutions go to infinity in finite time due to the non-lipschitz nature of the right hand side of (8).) From the models for the prey-predator, competition and mutualistic interactions, it is possible to derive some ecological consequences. In the prey-predator system, the prey brings advantage to the predator in the sense that its presence increases the number of predators at equilibrium, but the presence of predator decreases the equilibrium population of the prey. If the effect of the predator on the prey is too strong, predators consume all the preys, and, in the long time scale, predators lose advantage. For competition, the asymptotic equilibrium between the two species assumes lower values for both species when compared to the cases where they are isolated. In the mutualistic interaction, the situation is opposed to the competition case, where the equilibrium between the two species assumes higher values. However, for strong mutualistic interactions, we can have overcrowding as in the Malthusian model (1), leading to the death of the species by over consumption of resources. These conclusions, derived from the mathematical models (8) and (11), are in agreement with the biological knowledge about predation, competition and mutualism, [13] and [14]. Another model for the prey-predator interaction that has a conceptual and historical importance is the Lotka-Volterra model. This model has been used as an explanation to justify the resumption of carnivore fishes, after the cessation of fish-

15 14 R. Dilão ing in the North Adriatic Sea after the First World War, [8]. To be more specific, the prey-predator Lotka-Volterra interaction model is, { dx = r dt x x(1 c 1 y) dy dt = r (14) yy(c 2 x 1) where c 1, c 2, r x and r y are constants. This model obeys the prey-predator conditions in (10), but assumes that predators have an intrinsic negative growth rate and do not survive without preys. For preys alone, it assumes that they have exponential growth as in model (1). The Lotka-Voltrerra model (14) has one horizontal and one vertical nullcline in phase space, Fig. 6, and one non-zero fixed point with coordinates (x, y) = (1/c 2, 1/c 1 ). One of the eigenvalues of the Jacobian matrix of (14) calculated at the fixed point is zero, and as the conditions on the Hartman-Grobman theorem fail: the local structure of the flow cannot be characterized by the linear analysis around the fixed point. It can be shown that, in the first quadrant of phase space, the solution orbits of (14) are closed curves around the fixed point, Fig. 6, corresponding to oscillatory motion in the prey and predator time series (for a proof see [12]). Moreover, along each phase space cycle, the temporal means of prey and predators are independent of the amplitude of the cycles, being given by, x =1/c 2 and y =1/c 1, respectively. This property of the solutions of equations (14) has been used to assert that fluctuations in fisheries are periodic but the time average during each cycle is conserved, [8, p. 93]. Fig. 6. Qualitative structure of the flow in phase space of the Lotka-Volterra system of equations (14), for parameter values c 1 = c 2 = r x = r y = 1. Away from the non-zero fixed point, the solutions are periodic in time, suggesting a simple explanation for the oscillatory behaviour observed in preypredator real systems. It can be shown that the orbits of the system of equations (14) are the level sets of the function, H(x, y) =r y log x r yc 2 x + r x log y r xc 1 y. One of the important issues in the Lotka-Volterra model is to suggest the possibility of existence of time oscillations in prey-predator systems. A long-term ob-

16 Mathematical Models in Population Dynamics and Ecology 15 servation of prey-predator oscillations was provided by the hare-lynx catches data during 90 years, from the Hudson Bay Company, [14] and [15]. The catches of lynx and hare are in principle proportional to the abundances of these animals in nature, and the time series shows an out of phase oscillatory abundance, with the lynx maximum preceding the hare maximum. Making a naïf analogy between the solutions of the Lotka-Volterra model and the oscillations found in the lynx-hare interaction, it turns out that the maximum number of preys is observed before the maximum numbers of predator. This is in clear disagreement with the Lotka-Volterra model where the prey maximum precedes in time the predator maximum, Fig. 6. Several attempts were made to explain this out of phase behaviour but no consistent explanations have been found, [15]. One possible meaningful argument against the plausibility of the Lotka-Volterra model (14) to describe the prey-predator interaction is based on the property that any perturbation of the right hand side of equation (14) destroys the periodic orbits in phase space. In mathematical terms, it means that the Lotka-Volterra system (14) is not structurally stable or robust. In general, a two-dimensional dynamical system is structurally stable if all its fixed points obey the conditions of the Hartman- Grobman theorem, and there are no phase space orbits connecting unstable fixed points (saddle points), [11] and [12]. The only types of structurally stable twodimensional differential equations with periodic orbit in phase space are equations with isolated periodic orbits or limit cycles. In this case, the growth rate functions f and g must be at most quadratic, and several models with this property appeared in the literature, [12] and [15]. However, all these models show the same wrong out of phase effect as in the hare-lynx data. In modern theoretical ecology, the development of more specialized models relies on the conditions (10) and on further assumptions on the functional behaviour of the growth rate functions, [13], [12] and [14]. In some cases, the assumptions are introduced in analogy with some mechanisms derived from chemical kinetics, [6], [16] and [17]. For example, the mechanisms, A + x (1 r1 + e 1 )x x + y (1 r2 + e 2 )y + cx x d1 y d2 with e 1 > 0, e 2 > 0, r 1 > 0, r 2 > 0 and c<1, and, A + x (1 r1 + e 1 )x B + y (1 r2 + e 2 )y x + y r3 c 1 x + c 2 y x d1 y d2 with e 1 > 0, e 2 > 0, r 1 > 0, r 2 > 0, r 3 > 0, c 1 > 0 and c 2 > 0, are examples (15) (16)

17 16 R. Dilão of possible mechanisms for the prey-predator and generic biotic interactions. The phase space structure of the orbits of the Lotka-Volterra system (14) and the model (8)-(11) are different from the ones derived from the model equations associated to (15) and (16). However, the mechanistic interpretation of models (15) and (16) are closer to the biological situations. A detailed account of models for predation and parasitism is analysed in [18] and [19]. 3. Discrete models for single populations. Age-structured models One important fact about the individuals of a species is the existence of age classes and life stages. Within each age class, the individuals of a species behave differently, have different types of dependencies on the environment, have different resource needs, etc.. For example, in insets, three stages are generally identified: egg, larval and the adult. In mammals, in the childhood phase, reproduction is not possible, neither hunting nor predation. To describe a population with age classes or stages, we can adopt a discrete formalism, where the transition between different age classes or stages is described in matrix form. One of the advantages of this type of models is that they can be naturally related with field data. One discrete model that accounts for age or stage classes has been proposed by Leslie in 1945, [20]. The Leslie model considers that, at time n, a population is described by a vector of population numbers, (N n ) T =(N1 n,..., N m), n where Ni n is the number of individuals with age class i (or in life stage i). The time transition between age classes is described by the map, N n+1 = AN n (17) where A is the Leslie time transition matrix. Under the hypothesis that from time n to time n + 1, the individuals die out or change between consecutive age classes, the matrix A has the form, 0 e 2 e 3 e k 1 e k α A = 0 α (18) α k 1 0 where the e i are fertility coefficients, and the α i are the fraction of individuals that survive in the transition from age class i 1toi. Clearly, e i 0 and 0 <α i 1. We consider always that e k > 0, where e k is the last reproductive age class. If e k = 0, the determinant of matrix A is zero. Obviously, we can have populations with age classes such that e p = 0 and α p > 0, for p>k. In this case, if e k > 0, the solutions Np n, with p>k, are determined from the solutions obtained from the discrete difference equation (17) in dimension k. For example, if e k+1 = 0, then

18 Mathematical Models in Population Dynamics and Ecology 17 Nk+1 n = α kn n 1 k. Therefore, without loss of generality, we always consider that e k > 0 and e p = 0, for p>k. Fig. 7. a) Total Portuguese population distributed by age classes for 1991, 1992 and 1999, [21]. b) Probability of survival between age classes calculated with the population data of 1991 and The fraction of individuals that survive in the transition from age class j 1toj is given by, α j = N n+1 j+1 /N n j. In Fig. 7a), we show the distribution of age classes for the Portuguese population obtained from the census of 1991, 1992 and 1999, [21]. As it is clearly seen, the population in 1992 and 1999 is approximately obtained from the population in 1991 by a translation along the age axis, a property shared by the Leslie transition matrix (18). In Fig. 7b), we show the values of the survival probability α j as a function of the age classes, calculated from the census of For age classes j<45, the values of α j are close to 1. The solution of the Leslie discrete map model (17) (18) is easily determined. As the discrete equation (17) is linear, its general solution is, [22], Ni n = k c ijλ n j (19) j=1 where the c ij are constants determined by the initial conditions and the coefficients of the matrix A, and the λ j are the eigenvalues of (18). To simplify, we assume that the eigenvalues of A have multiplicity 1. As A is a non-negative matrix with non-zero determinant (e k > 0), by the Frobenius-Perron theory, [23], its dominant eigenvalue λ is positive with multiplicity 1, implying the existence of a non-zero steady state if, and only if, λ = 1. If, λ<1, the solutions (19) go to zero, as n. If, λ>1, the solutions (19) go to infinity. Calculating the characteristic polynomial of A, we obtain (by induction in k), k i P (λ) =( 1) k λ k e i λ k i (20) i=2 j=2 α j 1

19 18 R. Dilão Imposing the condition that λ = 1 is a root of the polynomial (20), from the condition P (1) = 0, we define the constant, G := k i=2 e i i α j 1 (21) where G is the inherent net reproductive number of the population. If we make the approximation, α j 1, we have the net fertility number G := k e i. i=2 The condition for the existence of asymptotic stable population numbers, given by the dominant eigenvalue of A, can be stated through the inherent net reproductive number of the population. So, in a population where G<1, any initial condition leads to extinction. If, G>1, we have unbounded growth. If, G =1,in the limit n, the population attains a stable age distribution. The Leslie model is important to describe populations where there exists a complete knowledge of the life cycle of the species, including survival probabilities and fertilities by age classes. For example, in the Leslie paper [20], it is described a laboratory observation of the growth of Rattus norvegicus. For a period of 30 days, the projected total population number was overestimated with an error of 0.06% of the total population. For human populations, survival probabilities are easily estimated from census data, Fig. 7. However, data from the fertility coefficients are difficult to estimate due to sex distinction and to the distribution of fertility across age classes. For an exhaustive account about Leslie type models, its modifications, and several case studies we refer to [23], [24] and [13]. For tables of the world population by country and the measured parameters of the Leslie matrix, we refer to [25]. Comparing the discrete and the continuous time approaches, the Leslie population growth model presents exactly the same type of unbounded growth as the exponential model (1). To overcome the exponential type of growth, we can adopt two different points of view. One approach is to introduce a dependence of the growth rates on the population numbers, as it has been done in section 2, in the derivation of the logistic equation from the Malthusian growth equation. Another alternative is to introduce a limitation on the growth rates through the resource consumption of the population. These two types of development of the Leslie model lead to the introduction of the concepts of chaos and randomness and will be developed below. j=2 4. A case study with a simple linear discrete model Here, we introduce a simplified discrete linear model enabling to make projections about human population growth based on census data. With this simple model, we avoid the difficulty associated with the choice of the fertility coefficients by age classes, a characteristic of the Leslie model.

20 Mathematical Models in Population Dynamics and Ecology 19 We characterize a population in a finite territory at time n by a two-dimensional vector (B n,n n ) T, where B n represents the age class of new-borns, individuals with less than 1 year, and N n represents the total number of individuals with one or more years. By analogy with the Leslie approach of the previous section, the time evolution equations are now, ( ) ( )( ) B n+1 0 e B n N n+1 = βα N n (22) where e is the (mean) fertility coefficient of the population, α is the probability of survival of the total population between consecutive years, and β is the probability of survival of new-borns. In census data, the fertility coefficient is given in number of new-borns per thousand, but here we use the convention that the fertility coefficient is given in number of new-borns by individual. Following the same approach as in the previous section, the solution of the discrete equation (22) is, B n = c 1 λ n 1 + c 2 λ n 2 N n = c 3 λ n 1 + c 4 λ n 2 (23) where λ 1 and λ 2 are the eigenvalues of the matrix defined in (22), and the c i are constants to be determined from the initial data taken at some initial census time n 0. If the dominant eigenvalue of the matrix in (22) is λ = 1, in the limit n, the solution (23) converges to a non-zero constant solution, from any non-zero initial data. As the characteristic polynomial of the matrix in (22) is, P (λ) =λ 2 αλ eβ the condition of existence of a non-zero steady state is, I = α + eβ = 1 (24) As in (21), we say that I = α + eβ is the inherent net reproductive number of the population. If, I>1, then λ>1, and the solution (23) diverges to infinity as n. If, I<1, then the solution (23) goes to zero. In order to calibrate the simple model (22), we take the census data for the Portuguese population in the period , Fig. 8. As we see from Fig. 8, the total Portuguese population without newborns shows strong variations, sometimes with a negative growth rate. This negative growth rate is due to emigration, decrease of population fertility and other social factors. The data for new-borns also shows negative growth rates. Therefore, the growth behaviour shown in Fig. 8 is influenced by other factors that are necessary to quantify. The values of the parameters α, β and e, are calculated from the census data and are shown in Fig. 9. The probability of survival of the population is approximately constant with mean α =0.9891, and a standard deviation of the order of The coefficient of fertility e and the new-borns survival probability β vary along the

21 20 R. Dilão Fig. 8. Portuguese population and new-borns for the years , from reference [21]. years. The last two coefficients are very sensitive to socio-economic and technological factors, suggesting that, for growth predictions, we must introduce into model (22) their time variation. Fig. 9. a) Probability of survival α, and (b) fertility coefficient e for the period c) Probability of survival of new-borns β in the period The probabilities of survival α and β have been calculated with the death rate data by thousand habitants, [21]. In b) and c), we show the fitting functions (25), for the parameter values (26). From the data of Fig. 9, the net reproductive number can be estimated. In Fig. 10, we show the variation of I = α + eβ for the period For 1960, we have I = and, for 1999, I = , both very close to the steady state condition (24). Therefore, during this period of time, the Portuguese population is growing with a net reproductive number I>1, but very close to 1. The decrease in the population number in the period is essentially due to emigration. To make population growth projections, we consider that α is constant, Fig. 9, with the mean value α =0.9891, and we consider that e and β are time varying functions. Due to the form of the curves in Fig. 9, the functions, e(t) =c 1 + c 2 c 3 +(t 1945) c 4 β(t) =1 c 5 e c (25) 6(t 1960)

22 Mathematical Models in Population Dynamics and Ecology 21 are reasonable choices, with fitting constants, c 1 = ,c 2 = ,c 3 = ,c 4 = c 5 = ,c 6 = (26) In Fig. 9, we show the fitting functions (25), for the parameter values (26). In the limit t, the new-borns survival probability converges to 1 and the mean fertility coefficient converges to c , which corresponds roughly to 10 newborns per thousand individuals in the population. The census value of e for 1999 corresponds to 11.6 new-borns per thousand. Fig. 10. Dots represent the inherent net reproductive number of the Portuguese population, calculated from the data of Fig. 9 ( ). The two lines correspond to the two possible projections for the net reproductive number I, for the period In estimate a), we have considered that the time dependence of β and e is given by (25), for the parameter values (26). In estimate b), we have taken constant values for β and e, obtained with the 1999 census values, [21]. To estimate the population growth for the period , we adopt two strategies for the iteration of map (22). In the first case, we iterate (22) with the time dependent functions (25), and we introduce as initial conditions the census data for 1999, Fig. 11. In the second case, we take for β and e the 1999 values. We also apply these two strategies to estimate the net reproductive number (24) as a function of time, Fig. 10. With the simplified model (22), it is possible to make a short time projection of population numbers. However, for a good calibration and greater accuracy, emigration and immigration factors must be taken into account. From the data and the fits in Figs. 10 and 11, we can derive several conclusions. The projections for the period show two different growth behaviours: In case a) of Fig. 10 and 11, we have, B 2010 =99, 966 and N 2010 = 9,835,840, with B 1999 = 115, 440 and N 1999 = 9,882,150, implying a negative growth with an inherent net reproductive number I<1. In case b), we have, B 2010 = 115, 251 and